(Solved): Please show the details Eigenvalues and eigenfunctions: Find all the eigenv ...

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Eigenvalues and eigenfunctions: Find all the eigenvalues $${?}_n$$ and eigenfunctions $$X_n(x)$$ for the boundary value problem $$X^{??}+?X=0,X^{?}(0)=0,X^{?}(?)=0,$$ following the steps outlined in class (see Ch.2, pp.27-30 of the course notes); you may assume that $$??\mathbb{R}$$ (i.e. the eigenvalues $$?$$ are real-valued). These boundary conditions (fixing the value of the derivative $$X^{?}$$ ) are called Neumann BCs. In addition: - Use the result from Question 4 to show that $${?}_0^{?}{X}_m(x)X_n(x)dx=0$$ for $$m?=n$$ (this is called an orthogonality condition). - What is the value of this integral if $$m=n$$ ? (Consider all possible cases for integers $$m,n?0$$.)